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CORRELATIONS A Group Contribution Model for the Prediction of the Thermal Conductivity of Polymer Melts Chongli Zhong,* Qingyuan Yang, and Wenchuan Wang Department of Chemical Engineering, P.O. Box 100, Beijing University of Chemical Technology, Beijing 100029, China
A group contribution model was proposed for the prediction of the thermal conductivity of polymer melts. The model requires only the existing group contribution methods for the estimation of specific heats, densities, and melting temperatures in the calculation of thermal conductivity for polymer melts. For 11 polymer melt systems tested, predictive accuracy of the model is normally better than 10%. Because only group parameters are needed in the calculation, this model is predictive and is very convenient for practical use. Introduction Thermal conductivity is an important material property in many polymer processing operations, because heat needs to be added or removed for most polymer processing processes, and such processes cannot be properly controlled or optimized unless the heat transfer is well understood. Investigation of thermal conductivities of solid and molten polymers has been carried out by many researchers theoretically1-4 and experimentally.5-8 Datta and Mashelkar9 have summarized the experimental data, measurement techniques, correlations, and theory up through about 1983, and Caruthers et al.10 reviewed the existing data, theories, and correlations up to about 1998. Though a lot of efforts have been made, methods for accurately predicting the thermal conductivity for polymer systems are less than satisfying,10 and the existing correlations usually require that certain data be available in the calculations. For example, the correlation proposed by Luba et al.1 requires the density at 298 K, specific heats, and melting temperature. Van Krevelen3 has proposed a generalized curve for amorphous polymers, where the ratio λ(T)/λ(Tg) is correlated with the ratio T/Tg, where Tg is the glass temperature, and λ(T) and λ(Tg) are the thermal conductivities at T and Tg, respectively. However, unless Tg and λ(Tg) (or a λ(T) at a given T) were known, the method could not be used. Though van Krevelen3 has proposed a group contribution method for estimating the thermal conductivity at 298 K when no experimental thermal conductivity data are available, the generalized curve has to be used for the calculation of thermal conductivities at other temperatures, which is inconvenient, and, more important, the errors are * Correspondingauthor. E-mail:
[email protected]. Fax: +86-10-64436781. Tel.: +86-10-64419862.
relatively large. It seems that, to our best knowledge, no reliable predictive models are available at this moment. Therefore, a predictive model providing good predictions for the thermal conductivity of polymer melts for a wide temperature range is highly needed, which is the motivation of this work. Development of the Group Contribution Model Luba et al.1 proposed a correlation for the thermal conductivity of polymer melts as follows:
λ)
1.2 × 10-2CpF1.33 Tm0.216M0.300
(1)
where λ is the thermal conductivity, Tm is the melting temperature, M is the mer weight, Cp is the specific heat of the melt at the temperature desired, and F is the polymer density at 298 K. Though the equation is empirical, it has a solid root in a number of theoretical sources.1 The model gives good estimations for thermal conductivity of polymer melts; however, as mentioned above, it requires densities at 298 K, specific heats at temperatures desired, and melting temperatures, which may not be available in some cases. Therefore, it is useful to introduce group contribution estimation methods for the above properties required in the model and to develop a group contribution model so that the thermal conductivities can be estimated based only on the chemical structure of the polymers and the relevant group parameters. Therefore, the commonly used group contribution methods for estimating Cp, F, and Tm are adopted, where many group parameters are readily available. Details of the group contribution
10.1021/ie000926f CCC: $20.00 © 2001 American Chemical Society Published on Web 08/23/2001
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Table 1. Group Parameters Used in This Work11 group
Cp/ (J/mol‚K)
Ym × 10-3/ (g‚K/mol)
Vνdw/ (cm3/mol)
-CH2-CH(CH3)-C(CH3)2-CH(C6H5)-C6H4-O-C(CH3)(COOCH3)-CO-CONH-
30.4 57.85 81.2 144.15 113.1 35.6 146.2 52.8 85
5.7 13.0 12.1 48 38 13.5 41.5 12 45
10.23 20.45 30.67 52.62 43.32 5 46.7 11.5 18.8
methods adopted in this work for polymer melts are as follows:11
Cp(T) ) Cp(298 K) [1 + 1.2 × 10-3(T - 298)] (2) Cp(298 K) )
∑i niCp,i(298 K)
(5) (6)
∑i niYm,i
(7)
M
where Cp,i(298 K) and VvdW(298 K) are the specific heat and van der Waals volume group parameters for group i at 298 K, respectively. Ym,i is the group parameter for the melting temperature, and ni is the number of group i in the mer. Cp, F, and Tm estimated by these equations are in J/mol‚K, g/cm3, and K, respectively. The group parameters can be found in the paper of van Krevelen,11 and those used in this work are listed in Table 1. With the above group contribution methods, the required Cp, F, and Tm can be estimated as long as the chemical structure of the mer is known. However, because all of the group contribution methods may have large uncertainties in the estimation of the relevant properties, the four universal constants in eq 1 need to
(8)
TmcMd
A total of 11 polymer melt systems were collected, and 8 of them were used to regress the four parameters in eq 8; the other three systems are not included in the regression, which will be used to test the predictive reliability of the proposed model later. The new constants obtained are shown in eq 9, which is the new group contribution model proposed in this work.
λ)
∑i niVνdW,i(298 K)
aCpFb
Results and Discussion
(4)
F ) M/V(T) Tm )
λ)
(3)
V(T) ) VνdW(298 K) (1.3 + 1 × 10-3T) VνdW(298 K) )
be reregressed with Cp, F, and Tm calculated from the adopted group contribution methods. Therefore, eq 1 can be rewritten as follows, and the constants will be regressed later.
6.833CpF0.793
(9)
Tm0.666M1.022
where λ, Cp, F, Tm, and M are in W/m‚K, J/mol‚K, g/cm3, K, and g/mol, respectively. The calculated results for eight polymer melts are shown in Table 2, which shows that eq 9 gives good estimations for the thermal conductivities of the eight polymer melts considered, with the average absolute deviations (AADs) being around 10%. Though it is a correlation in this case, because group contribution methods are used for calculating the properties required and the four parameters are universal constants for any polymer melts, the model has predictive capability. Taking this into account, the accuracy of the model can be thought to be satisfactory. Therefore, it is clear that, as long as the chemical structure of a polymer melt is known, the model can be used to predict the temperature dependence of the thermal conductivity with reasonable accuracy, which is very convenient and is of practical use. The parallel calculations were carried out using the van Krevelen method;3 the results are also reported in Table 2. It should be pointed out that the thermal conductivities other than 298 K were estimated using the generalized curve proposed by his work. Obviously,
Table 2. Correlated Results of the Thermal Conductivities for Eight Polymer Melts AADλa polymer
M h n × 10-3
no. of data points
temp range
this work
van Krevelen
data source
Nylon-6 Nylon-6,10 low-density polyethylene high-density polyethylene poly(ethylene glycol) poly(methyl methacrylate) polypropylene polystyrene
13.0 18.0 12.3 26.0 47.0 -
5 5 4 4 9 8 4 8
483-510 465.1-518.1 436.1-511.1 470.1-553.1 333-413 393-463 433.1-520.1 393-463
7.9 8.7 9.1 9.0 9.4 13.3 7.2 1.6
38.9 31.5 46.7b c 17.6 10.5 35.3 32.9
5 7 7 7 6 5 7 5
AADλ ) (1/N)∑ |λi - λi |/λiexp × 100. b Only one data point whose temperature is within the T/Tg range of the generalized curve. c No data points whose temperature is within the T/Tg range of the generalized curve. calc
a
exp
Table 3. Predicted Results of the Thermal Conductivities for Three Polymer Melts AADλa M hn×
polymer polyoxymethylene Nylon-6,6 polycarbonate a
10-3
13.1
AADλ ) (1/N)∑ |λi - λi |/λiexp × 100. calc
exp
no. of data points
temp range
this work
van Krevelen
data source
2 1 5
473.1-493.1 523 466.8-548.1
9.2 1.7 14.9
37.6 12.6 39.0
8 12 13
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adopted here are widely used and many group parameters are readily available, the model has a good predictive perspective and can be used for many kinds of polymers. The calculated results show that the model is reliable and the predictive accuracy is better than 10% for most systems. Therefore, the model is of practical use. A similar model is under development for polymers below the melting temperature, which will be reported in our upcoming paper. Acknowledgment Financial support of the Ministry of Education of China (Contract 99013) and the State Key Fundamental Research Plan (No. G2000048) is greatly appreciated. Literature Cited Figure 1. Calculated and experimental thermal conductivities of Nylon-6,10 melt.
the model proposed in this work gives much better accuracy. Furthermore, the proposed method is much simpler than the van Krevelen method. The calculated temperature dependence of the thermal conductivities of Nylon-6,10 melt is shown in Figure 1, where the experimental data are also reported. Though the model predicts larger values than the experimental data, it gives the same trend as the experimental observation. To test the predictive capability and reliability of the proposed model, it was further used to predict the thermal conductivities of three other polymer melts, which were not included in the previous regression of the parameters. The predicted results are reported in Table 3, which shows that the proposed model gives accuracy similar to those in Table 2. This indicates that the model is reliable and can be used for the prediction of the thermal conductivity of polymer melts. The results from the van Krevelen method are also listed in Table 3, which shows accuracy similar to that in Table 2. Thermal conductivity data of polymer blends and copolymers are practically interested; therefore, the model is more useful if it can be applied to these systems. To our experience, the proposed model may be applicable to polymer blends and copolymers. We are collecting data from literature, and when the data are enough, the model will be further tested to get a definite conclusion. Conclusion The proposed model can give good predictions for the thermal conductivity of polymer melts based only on the information of the chemical structure of the polymers considered. Because the group contribution methods
(1) Luba, M.; Pelt, T.; Griskey, R. G. Correlating and Predicting Polymer Thermal Conductivities. J. Appl. Polym. Sci. 1979, 23, 55. (2) Saeki, S.; Tsubokawa, M.; Yamaguchi, T. Correlation Between the Equation of State and the Temperature and Pressure Dependence of Thermal Conductivity of Polymers and Simple Liquids. Polymer 1990, 31, 1919. (3) Van Krevelen, D. W. Properties of Polymers. Their Correlation with Chemical Structure, Their Numerical Estimation and Prediction from Additive Group Contributions, 3rd rev. ed.; Elsevier: Amsterdam, The Netherlands, 1990. (4) Dashora, P.; Gupta, G. On the Temperature Dependence of the Thermal Conductivity of Linear Amorphous Polymers. Polymer 1996, 37, 231. (5) von Lohe, P. Heat conductivity of high-polymer melts. I. Poly(methyl methacrylate), polystyrene, Nylon 6, and poly(3,3bis(chloromethyl)oxacyclobutane). Kolloid-Z. 1965, 203, 115. (6) von Lohe, P. Heat conductivity of high-polymer melts. II. Influence of the degree of polymerization on the thermal conductivity. Kolloid-Z. 1965, 204, 7. (7) Fuller, T. R.; Fricke, A. L. Thermal Conductivity of Polymer Melts. J. Appl. Polym. Sci. 1971, 15, 1729. (8) Keating, M. Y.; Aclaren, C. S. Thermal Conductivity of Polymer Melts. Thermochim. Acta 1990, 166, 69. (9) Datta, A.; Mashelkar, R. A. In Transport Phenomena in Polymeric Systems; Mashelkar, R. A., Majumdar, A. S., Kamal, R., Eds.; Wiley Eastern: New Delhi, India, 1987 (Part I), 1989 (Part II). (10) Caruthers, J. M.; Chao, K.-C.; Venkatasubramanian, V.; Sy-Siong-Kiao, R.; Novenario, C. R.; Sundaram, A. Handbook of Diffusion and Thermal Properties of Polymers and Polymer Solutions; DIPPR: New York, 1998. (11) Van Krevelen, D. W. Group Contribution Techniques for Correlating Polymer Properties and Chemical Structure. In Computational Modeling of Polymers; Bicerano, J., Ed.; Marcel Dekker: New York, 1992. (12) Mark, J. E. Physical Properties of Polymers Handbook; AIP Press: New York, 1996. (13) Dietz, W. The thermal conductivity and diffusivity of polymers. Colloid Polym. Sci. 1977, 255, 755.
Received for review October 27, 2000 Revised manuscript received February 15, 2001 Accepted July 2, 2001 IE000926F
